What is an Earthquake?

What is an Earthquake?

Natural Seismicity

Induced Seismicity

Physics Recap

Cats on a Roof

What is a Fault?

Image Source - Wikimedia

Earthquakes as Fault Slip

What provides the driving force?

How do we measure earthquakes?

How do we locate earthquakes? (1)

How do we locate earthquakes? (2)

How do we locate earthquakes? (3)

How do we locate earthquakes? (4)

Earthquake Catalogues

• x Easting / Longitude
• y Northing / Latitude
• z Depth (often nominal)
• t Time (YYYY-MM-DD HH:MM:SS)
• m Magnitude

A note on Magnitudes

Magnitudes are measured on logarithmic scales and typically reported to 1 decimal place.

\[\begin{align*} \text{Richter Scale} \rightarrow \text{Local Magnitude} &\sim \text{Amplitude}. \\ &\text{Moment Magnitude} %\sim \text{Energy}. \end{align*}\]

\[ E \propto A^{1.5} \text{ so } 10\times \text{ or } 32\times \text{ increase.}\]

What is an Aftershock?

What is an Aftershock?

Locations: Point Processes

Locations: Inhomogeneous Poisson Processes

Introduce an intensity function \(\lambda(a): A \rightarrow \mathbb{R}^+_0\).

\[ N(A) \sim \text{Pois}\left(\Lambda(A)\right) \quad \text{where} \quad \Lambda(A) = \int_A \lambda(a) \mathrm{d}a.\]

Then

\[ X_i \overset{\text{i.i.d}}{\sim} f_X(a) = \frac{\lambda(a)}{\Lambda(a)}.\]

Physically motivated forms for \(\lambda(a)\), e.g. Bourne and Oates (2018): \[\Lambda \propto \exp(b_0 + b_1 z(a))\].

Locations: Adding in Aftershocks

Hawkes Processes add self excitation.

\[\begin{equation*} \lambda(t; \mathcal{H}_t) = \mu + \sum_{i: t_i < t} \alpha \exp\{-\beta(t - t_i)\}. \end{equation*}\]

Locations: Adding in Aftershocks - a better way

\[\begin{align*} \lambda(t) &= \mu + \sum_{i: t_i < t} \alpha \exp\{-\beta(t - t_i)\} \\ &= \mu + \sum_{i: t_i < t} \frac{\alpha}{\beta} \beta \exp\{-\beta(t - t_i)\} \\ &= \mu + \sum_{i: t_i < t} \alpha^\prime \beta \exp\{-\beta(t - t_i)\} \\ &= \mu + \sum_{i: t_i < t} \kappa(\cdot; \alpha^\prime) h(t - t_i; \beta)\}. \end{align*}\]

What about magnitudes?

Biased estimation if rounding ignored.

All Together Now: ETAS model

\[ \lambda(t;\mathcal{H}_t, \theta) = \mu + \sum_{i:t_i < t} \kappa(m_i; K,a) h(t-t_i; c, p).\]

\[ \lambda(t;\mathcal{H}_t, \theta) = \mu + \sum_{i:t_i < t} K e^{a(m_i - m_c)} (p-1)c^{p-1} \left(1 + \frac{t-t_i}{c}\right)^{-p}.\]

\[ M_i \overset{\text{i.i.d}}{\sim} \text{Exp}(\beta). \]

ETAS Extensions

ETAS Model Concerns

Veen and Schoenberg (2012) highlight several issues with the ETAS likelihood:

ETAS as a Branching Process

Conditional Inference

Conditioning on \(B\) provides very simple conditional distributions:

\[\begin{align*} \mu &| K, a, c, p, B - \text{Homogeneous PP}\\ K, a &| \mu, c, p, B - \text{Poisson Regression} \\ c, p &| \mu, K, a, B - \text{Power-law} \\ B &| \mu, K, a, c, p - B_i \text{ multinomial}. \end{align*}\]

Unified Model and Dependence

Following from branching process representation Varty (2021) considers:

Selecting M_c

Varty et al (2021) and Murphy et al (2023)

Errors in EV and PP models

There are an awful lot of measurement errors being ignored here… (Yue 2023-2025)

Wrapping Up

1. Earthquakes are significant natural hazards but have received relatively little attention.

2. State of the art models are reasonably accessible.

3. Lots of contributions that could be made to extend these models by statistically minded folks.